Yes, your right love, it is an awful mess. I've being stuck in the middle of two similar cases in the past - my fathers and 6 months later my mothers passing. & I know only too well what it's like when others step in to push you out.
I take it that your brother and his wife were not divorced and that is why she can give instructions to the coroner? Best sometimes to step back and let them get on with it. You need to be there for your Mum at the moment as I am sure she is brokenheart as it is without all that nonesense going on.
The wife will HAVE to register the death, because she will need the certificate for the burial. And even if she makes all the arrangments, she cannot stop your family from attending. Just keep your dignity for the sake of your brother and his wee girl. You have a hard time in-front of you and all you can do is bite your tongue and pray.
As you say, your not in a position to pay out the cost of the funeral, so let her get on with it, the last thing you need is to be left with the bill and then having to try and claim it back from your brothers estate, which would be a long process.
I hope that the ex will respect any wishes you have, afterall, he was a son and a brother long before he was a husband.